Page:4SIGHT manual- a computer program for modelling degradation of underground low level waste concrete vaults (IA 4sightmanualcomp5612snyd).pdf/141

65. Update Pressures.

The pressures are updated using the continuity equation:

$${\frac {\partial {p}}{\partial {t}}}=\nabla \cdot \rho \mathbf {v}$$

where $\mathbf {v}$ the intrinisic pore fluid velocity and $\rho$ is the pore fluid density. An equation for the bulk material can be obtained from a volume average over a represetative volume V

For these equations, assume $\rho$ is a constant since water is virtually incompressible. Rearranging and simplifying the above equation gives:

$${\frac {\partial {}}{\partial {t}}}{\frac {1}{V}}\int _{v}{\frac {\partial {p}}{\partial {t}}}d^{3}\mathbf {x} =-{\frac {1}{V}}\int _{v}\nabla \cdot \rho \mathbf {v} d^{3}\mathbf {x}$$

Since $\rho$ is zero outside the pore volume, if $V_{p}$ represents only the pore volume then the above equation simplifies to

$${\frac {\partial {\phi }}{\partial {t}}}-\nabla \cdot {\frac {\phi }{V_{p}}}\int _{V_{p}}\mathbf {v} d^{3}\mathbf {x}$$

which finally gives

$${\frac {\partial {\phi }}{\partial {t}}}=-\nabla \cdot \phi \mathbf {u} =-\nabla \cdot \phi \mathbf {v} _{D}$$

where $\mathbf {v} _{D}$ is the Darcy velocity. Substituting for the Darcy velocity gives

$${\frac {\partial {\phi }}{\partial {t}}}=-\nabla \cdot {\frac {k}{\mu }}\nabla p$$

⟨Function declarations 23⟩ $+\equiv$
 void update.pressures (void);

66. update.pressures

 void update.pressures()
 {
   int k, iteration;
   real perm [NUM_CELLS], imp [NUM.SURFACES], tol = 0.0005, $\varepsilon$;
   for (k = FIRST.CELL; k < NUM.CELLS; k++) perm[k]=$\kappa$[k] * CUB($\xi$[ke]/$\vartheta _{0}$ );
   $\Phi$[FIRST.CELL] = Pout + (NUM.CELLS - 1 + A A [FIRST.CELL]) * dP
   
   $\Phi$[NUM.SURFACES-1]=0;
   iteration = 0;
   do {
     for (k= FIRST.CELL + 1; k < NUM.SURFACES - 1; k++) {
       imp[k]= perm * $\Psi$[k + 1]/$\Delta X$[k] + perm[k -1]*$\Psi$[k-1]/$\Delta X$[k-1];
 /* **** tmp[k]-=0.5*(DX[k-1]_DX[k])*(PHIn[k]-PHIn[k-1])/DT; *****/
 imp[k]*=1.0/(perm[k]/$\Delta X$[k] = perm[k-1]/$\Delta X$[k-1]);
 }
 $\varepsilon$=0.0;
 for (k= FIRST_CELL+1; K<NUM_SURFACES-1;k++) {
   if($\Psi$[k]>0.0 $\epsilon$=MAX($\epsilon$,$\Psi$[k] -imp[k])/$\Psi$[k]);